Critical SK autocorrelation processes and dynamics across the temperature transition

The explanation

Take n tiny magnets, called spins, each pointing up or down. Every pair is joined by a random strength, positive or negative. This is the Sherrington–Kirkpatrick (SK) model, introduced in 1975Source 1. A number β, the inverse temperature, sets how strongly the spins feel those pulls. At β = 1 the model changes phase. Below 1 it behaves like a hot, loose system. Above 1 it becomes a spin glass, frozen in a rugged landscape.

Now let the spins move. Each spin carries a clock that rings on average once per time unit. When it rings, the spin is redrawn at random, given its neighbors. This is heat-bath dynamics. The mixing time is how long it takes, from the worst start, to look like a typical equilibrium sample.

The papers find three regimes. Take n = 1,000 spins. At β = 1, the mixing time grows like n^(2/3), here about 100 time units. That is about 100,000 single-spin updates, since each time unit has n of them. Below β = 1, mixing takes only about log n time units, here about 6.9 times a constant. Above β = 1, typical starts take time that grows faster than any power of n.

Earlier proofs covered only part of the hot phase. Bauerschmidt and Bodineau proved a key inequality for β below 1/4Source 2. Others turned results like this into fast mixing in that range. Later work pushed the range up to 1/2, and slightly past it at zero fieldSource 3. None of these bounds reach β = 1 by taking a limitSource 4. Physicists had seen the n^(2/3) scale in simulationsSource 5.

The spectral-gap paper proves that the dynamics stays well-mixed at every β below 1Source 6. Equivalently, the equilibrium law satisfies a Poincaré inequality, a bound on how much any function can vary, that does not grow with nSource 7. A second paper claims cutoff for every β below 1Source 8. Cutoff means the distance from equilibrium stays near its maximum and then drops suddenly, at time log n / (2λ(β)) for an explicit rate λ(β).

At β = 1, one paper proves that typical equilibrium starts are still far from equilibrium before time n^(2/3)Source 9. The critical mixing paper claims the matching upper bound, so the mixing time is n^(2/3 + o(1))Source 10. The o(1) means the exponent is right up to any small slack. It does not claim a cutoff or a leading constantSource 11. The obstacle at β = 1 is that equilibrium fluctuations are much larger than for independent spinsSource 12. Every bound must hold from every start and for every test function.

Above β = 1, one paper proves a barrier for typical starts. Starting from a sampled equilibrium state, the dynamics is still far from equilibrium at time exp(n^(1/10000))Source 13. That bound only matters for huge n. For a million spins it is about 2.72 time units. Another paper claims an upper bound for typical starts, below exp(n) by a tiny powerSource 14.

Two more papers look at β = 1 more closely. They scale time by n^(2/3) and multiply the average spin autocorrelation by n^(1/3)Source 15. With n = 1,000 that factor is 10. The limit is the same for Gaussian and ±1 strengthsSource 16. It stays random from sample to sampleSource 17. One paper starts from independent fair spins, a sudden cooling called a quenchSource 18. Its limit relaxes to the equilibrium one over long waiting timesSource 19.

How sure is this? Lean, a computer proof checker, has formal versions of several pieces. They include the spectral gap for every β below 1, and cutoff for β below 1/2. At β = 1, Lean bounds the mixing time between n^(2/3−ε) and e^(εn), so the matching upper bound is not in it. It also covers the β > 1 barrier. The universality papers and the β > 1 upper bound are not covered. No specialist review of these papers has been published.

What it doesn't show. This does not give a cutoff or a leading constant at the critical point.